Non-uniform ACC Circuit Lower Bounds

Abstract

The class ACC consists of circuit families with constant depth over unbounded fan-in AND, OR, NOT, and MOD$m$ gates, where $m > 1$ is an arbitrary constant. We prove:- $NTIME[2^n]$ does not have non-uniform ACC circuits of polynomial size. The size lower bound can be strengthened to quasi-polynomials and other less natural functions.- $E^{NP}$, the class of languages recognized in $2^{O(n)}$ time with an $NP$ oracle, doesn't have non-uniform ACC circuits of $2^{n^{o(1)}}$ size. The lower bound gives a size-depth tradeoff: for every $d$, $m$ there is a $\delta > 0$ such that $E^{NP}$ doesn't have depth-$d$ ACC circuits of size $2^{n^{\delta}}$ with MOD$m$ gates.Previously, it was not known whether $EXP^{NP}$ had depth-3 polynomial size circuits made out of only MOD6 gates. The high-level strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms can be applied to obtain the above lower bounds.